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In
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
, the chirp mass of a compact binary system determines the
leading-order The leading-order terms (or leading-order corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude.J.K.Hunter, ''Asymptotic Analysis and Singular Perturbation Theory'', 2004. http://www.m ...
orbital evolution of the system as a result of energy loss from emitting
gravitational waves Gravitational waves are oscillations of the gravitational field that travel through space at the speed of light; they are generated by the relative motion of gravitating masses. They were proposed by Oliver Heaviside in 1893 and then later by H ...
. Because the gravitational wave frequency is determined by orbital frequency, the chirp mass also determines the frequency evolution of the gravitational wave signal emitted during a binary's inspiral phase. In gravitational wave data analysis, it is easier to measure the chirp mass than the two component masses alone.


Definition from component masses

A two-body system with component masses m_1 and m_2 has a chirp mass of :\mathcal=\frac The chirp mass may also be expressed in terms of the total mass of the system M = m_1 + m_2 and other common mass parameters: * the reduced mass \mu = \frac: *: \mathcal = \mu^ M^, * the mass ratio q = m_1/m_2: *: \mathcal = \left \frac \right M, or * the symmetric mass ratio \eta = \frac = \frac\mu M = \frac = \left( \frac \right)^2: *: \mathcal = \eta^ M. *: The symmetric mass ratio reaches its maximum value \eta = \frac 14 when m_1 = m_2, and thus \mathcal = (1/4)^ M \approx 0.435\,M. * the geometric mean of the component masses m_ = \sqrt: *: \mathcal = m_ \left(\frac\right)^, *: If the two component masses are roughly similar, then the latter factor is close to (1/2)^ = 0.871, so \mathcal \approx 0.871 \, m_ . This multiplier decreases for unequal component masses but quite slowly. E.g. for a 3:1 mass ratio it becomes \mathcal = 0.846\,m_, while for a 10:1 mass ratio it is \mathcal = 0.779\,m_.


Orbital evolution

In
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, the phase evolution of a binary orbit can be computed using a
post-Newtonian expansion In general relativity, post-Newtonian expansions (PN expansions) are used for finding an approximate solution of Einstein field equations for the metric tensor (general relativity), metric tensor. The approximations are expanded in small paramet ...
, a perturbative expansion in powers of the orbital velocity v/c. The first order gravitational wave frequency, f, evolution is described by the differential equation :\frac=\frac\pi^\left(\frac\right)^f^, where c and G are the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
and Newton's gravitational constant, respectively. If one is able to measure both the frequency f and frequency derivative \dot of a gravitational wave signal, the chirp mass can be determined. To disentangle the individual component masses in the system one must additionally measure higher order terms in the post-Newtonian expansion.


Mass-redshift degeneracy

One limitation of the chirp mass is that it is affected by
redshift In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and increase in frequency and e ...
; what is actually derived from the observed gravitational waveform is the product : \mathcal_o = \mathcal(1+z) where z is the redshift. This redshifted chirp mass is larger than the source chirp mass, and can only be converted to a source chirp mass by finding the redshift z. This is usually resolved by using the observed amplitude to find the chirp mass divided by distance, and solving both equations using Hubble's law to compute the relationship between distance and redshift. Xian Chen has pointed out that this assumes non-cosmological redshifts ( peculiar velocity and
gravitational redshift In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well lose energy. This loss of energy correspo ...
) are negligible, and questions this assumption. If a binary pair of stellar-mass black holes merge while closely orbiting a
supermassive black hole A supermassive black hole (SMBH or sometimes SBH) is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions, of times the mass of the Sun (). Black holes are a class of astronomical ...
(an extreme mass ratio inspiral), the observed gravitational wave would experience significant gravitational and doppler redshift, leading to a falsely low redshift estimate, and therefore a falsely high mass. He suggests that there are plausible reasons to suspect that the SMBH's accretion disc and
tidal force The tidal force or tide-generating force is the difference in gravitational attraction between different points in a gravitational field, causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is the ...
s would enhance the merger rate of black hole binaries near it, and the consequent falsely high mass estimates would explain the unexpectedly large masses of observed black hole mergers. (The question would be best resolved by a lower-frequency gravitational wave detector such as LISA which could observe the extreme mass ratio inspiral waveform.)


See also

* Reduced mass * Two-body problem in general relativity


Note


References

{{Gravitational waves - Gravitational-wave astronomy